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Express-the-function-f-z-ze-iz-in-polar-form-and-separate-it-into-Real-and-Imaginary-part-M-m-

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Question Number 180159 by Mastermind last updated on 08/Nov/22
Express the function f(z)=ze^(iz) in polar form and separate it into Real and Imaginary part. M.m
$$\mathrm{Express}\:\mathrm{the}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{z}\right)=\mathrm{ze}^{\mathrm{iz}} \:\mathrm{in}\:\mathrm{polar} \\ $$$$\mathrm{form}\:\mathrm{and}\:\mathrm{separate}\:\mathrm{it}\:\mathrm{into}\:\mathrm{Real}\:\mathrm{and}\: \\ $$$$\mathrm{Imaginary}\:\mathrm{part}. \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$
Answered by Frix last updated on 08/Nov/22
z∈C z=a+bi f(z)=(a+bi)e^(−b+ai) =(a+bi)e^(−b) e^(ai) =(a/e^b )e^(ai) +((bi)/e^b )e^(ai) = =(a/e^b )(cos a +i sin a)+((bi)/e^b )(cos a +i sin a)= =(a/e^b )(cos a +i sin a)+(b/e^b )(−sin a +i cos a)= =((acos a −bsin a)/e^b )+((asin a +bcos a)/e^b )i
$${z}\in\mathbb{C} \\ $$$${z}={a}+{b}\mathrm{i} \\ $$$${f}\left({z}\right)=\left({a}+{b}\mathrm{i}\right)\mathrm{e}^{−{b}+{a}\mathrm{i}} =\left({a}+{b}\mathrm{i}\right)\mathrm{e}^{−{b}} \mathrm{e}^{{a}\mathrm{i}} =\frac{{a}}{\mathrm{e}^{{b}} }\mathrm{e}^{{a}\mathrm{i}} +\frac{{b}\mathrm{i}}{\mathrm{e}^{{b}} }\mathrm{e}^{{a}\mathrm{i}} = \\ $$$$=\frac{{a}}{\mathrm{e}^{{b}} }\left(\mathrm{cos}\:{a}\:+\mathrm{i}\:\mathrm{sin}\:{a}\right)+\frac{{b}\mathrm{i}}{\mathrm{e}^{{b}} }\left(\mathrm{cos}\:{a}\:+\mathrm{i}\:\mathrm{sin}\:{a}\right)= \\ $$$$=\frac{{a}}{\mathrm{e}^{{b}} }\left(\mathrm{cos}\:{a}\:+\mathrm{i}\:\mathrm{sin}\:{a}\right)+\frac{{b}}{\mathrm{e}^{{b}} }\left(−\mathrm{sin}\:{a}\:+\mathrm{i}\:\mathrm{cos}\:{a}\right)= \\ $$$$=\frac{{a}\mathrm{cos}\:{a}\:−{b}\mathrm{sin}\:{a}}{\mathrm{e}^{{b}} }+\frac{{a}\mathrm{sin}\:{a}\:+{b}\mathrm{cos}\:{a}}{\mathrm{e}^{{b}} }\mathrm{i} \\ $$
Commented by Mastermind last updated on 09/Nov/22
Thank you man
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{man} \\ $$
Commented by Mastermind last updated on 09/Nov/22
But in polar form
$$\mathrm{But}\:\mathrm{in}\:\mathrm{polar}\:\mathrm{form} \\ $$
Commented by Frix last updated on 09/Nov/22
ze^(iz) is the polar form and in polar form you cannot seperate the real and imaginary parts
$${z}\mathrm{e}^{\mathrm{i}{z}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{polar}\:\mathrm{form} \\ $$$$\mathrm{and}\:\mathrm{in}\:\mathrm{polar}\:\mathrm{form}\:\mathrm{you}\:\mathrm{cannot}\:\mathrm{seperate}\:\mathrm{the} \\ $$$$\mathrm{real}\:\mathrm{and}\:\mathrm{imaginary}\:\mathrm{parts} \\ $$

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